Arithmetic Geometric Progression

A sequence of the form S = a + (a+d)r + (a+2d)r2 + (a+3d)r3 + …. is called an arithmetic geometric progression.

Here, each term is the product of corresponding terms in an arithmetic and geometric series.

Tn = [a + (n-1) d]rn-1 is the nth term of A.G.P.

Sum of n terms of A.G.P. is given by S = a/(1-r) + dr(1-rn-1/(1-r)2) – [a+(n-1)d]rn/(1-r), where r is the common ratio, d is the common difference and a is the first term.

S∞ = a/(1-r) + dr/(1-r)2.

Harmonic Progression and Harmonic Mean

Let a, b and c form an H.P. Then 1/a, 1/b and 1/c form an A.P.

If a, b and c are in H.P. then 2/b = 1/a + 1/c, which can be simplified as b = 2ac/(a+c).

If ‘a’ and ‘b’ are two non-zero numbers then the sequence a, H, b is a H.P.

The n numbers H1, H2, ……,Hn are said to be harmonic means between a and b, if a, H1, H2 ……, Hn, b are in H.P. i.e. if 1/a, 1/H1, 1/H2, ..., 1/Hn, 1/b are in A.P. Let d be the common difference of the A.P., Then 1/b = 1/a + (n+1) d ⇒ d = a–b/(n+1)ab.

As the nth term of an A.P is given by an = a + (n-1)d, So the nth term of an H.P is given by 1/ [a + (n -1) d].

If we have a set of weights w1, w2, …. , wn associated with the set of values x1, x2, …. , xn, then the weighted harmonic mean is defined as

Questions on Harmonic Progression are generally solved by first converting them into those of Arithmetic Progression.

If ‘a’ and ‘b’ are two positive real numbers then A.M x H.M = G.M2

The relation between the three means is defined as A.M > G.M > H.M.

If we need to find three numbers in a H.P. then they should be assumed as 1/a–d, 1/a, 1/a+d

Four convenient numbers in H.P. are 1/a–3d, 1/a–d, 1/a+d, 1/a+3d

Five convenient numbers in H.P. are 1/a–2d, 1/a–d, 1/a, 1/a+d, 1/a+2d

Miscellaneous Sequences

Sequences containing Σn, Σn2, Σn3

? Σn = n(n+1)/2

Σn2 = n(n+1)(2n+1)/6

Σn3 = [n(n+1)/2]2

Using method of difference:

If T1, T2,T3 are the terms of sequence then the terms T2 – T1, T3 – T2, T4 – T3 …..may at times be in A.P. or sometimes may be in G.P. In such series, we first compute their nth term and then compute the sum to n terms, using sigma notation.

A series in which each term is composed of the reciprocal of the product of r factors in A.P., the first factor of the several terms being in the same A.P., then such series are solved by splitting the nth term as a difference of two.